Problem: Determine how many solutions exist for the system of equations. ${-x+y = -4}$ ${-18x+3y = 12}$
Convert both equations to slope-intercept form: ${-x+y = -4}$ $-x{+x} + y = -4{+x}$ $y = -4+x$ ${y = x-4}$ ${-18x+3y = 12}$ $-18x{+18x} + 3y = 12{+18x}$ $3y = 12+18x$ $y = 4+6x$ ${y = 6x+4}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x-4}$ ${y = 6x+4}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.